The following is uneditied !!
Chapters 10, 11 & 12 are okay.
7.3. Comparison of Boole's and Jevons's systems.
It was Jevons's belief that Boole went wrong in looking
at as a branch of Mathematical Analysis and that
his errors were the outcome of this. Every line in the
inciting out of a question in logic has to have a sense
from the point of view of logic. Boole lets a great number
of lines be used though they have no such sense.
Further, by Boole's account of the operation of addition
(+) of classes the operation may be done only on
classes which have no common elements, however, by
his rules he has to let ' x+x ' and ' x+(x+y) ' be parts of
equations though x and x, and x and x-1-y, have common
elements (if x and y are not Nothing). Again, while
it is a general law of Boole's algebra that xx (y+8)=
(x x y)-l-(xxa), it is possible for only one side of this
equation to be well-formed, given Boole's account of
+, because sometimes it is possible for the operation of
addition to be done on xxy and xxx, these two classes
having no common elements, while it is not possible
for the operation to be done on y and z,these two classes
having common elements, as when 2 and 3 are the
elements of x, 2, 4 and 5 are the elements of y and 3, 4
and 5 are the elements of z. Jevons put forward with
much force the suggestion, which had been made before
by De Morgan in connection with classes, that it
would be better if the operation of addidon in logic
were made to be one for which there is no need of the
things on which the operation is to be done to be completely
different and to have nothing whatever common
to them. In class algebra x-4-y would then be the class of
all the things that are elements of x or of y or of x and
of y. One value of this suggestion is its causing number
signs such as 2x(=x+x), which have no 8ense in logic,
to be droppedout of the algebra, x+x now being equal
to x. With this change of addition Boole's operation of
taking away (-) is no longer the opposite of + and i~
no longer like the ._ of arithmetic or number algebra ;
-x is made to be the class of all the things that are
elements of All but are not elements of x, so now, for
every class x, it will be true that x-4--x= 1, 1 being All,
unlike in number mathematics where .¢+-x=0.
- 47 - Another
value of Jevons's suggestion is that De Morgan's
Laws, put forward by De Morgan for classes, these
Laws being that -(x+y)=-xx -y and -{xxy)=--x+
-y, become true in the new theory, and this is a help in
the answering of questions about reasoning.
Though ]evons's theory of deduction was better in
some ways than Boole's, it was generally less good and
more complex. The effect of this was that only two of
his ideas were taken into the body of Mathematical
Logic by later workers. One was the idea of changing
the operation of addition. The other was his way of
making an expansion of an equation to take more qualities
or classes into account; for example, an expansion
of x=x is possible by which y is taken into account, as
in x=x=xx(y+-y)=(xxy)-I-(xx -y). Such expansions
were important for the later development of the
idea of what was named the ' normal form ' of a complex
of 8igns, the normal form being a fixed way of writing
any such complex so that comparisons between com-
plexes and decisions of certain sorts about their proper-
des may be readily made.
7.4. Peirce's points of agreeement with Boole and Jevons.
Charles Sanders Peirce (1839-1914) was a
teacher of philosophy, in logic, for only 5 years, at
Iohns Hopkins University 'm 1879-1884. He made his
living for 28 years, from the time he got his degree from
Harvard till 1887 when he was 48, from a branch of
United States government work in science; after 1887
he had no regular income and was often so poor and in
debt as not to have enough money for food and heating.
His ideas on Mathematical Logic were given in
papers, 011 recorded in private notebooks, whose writing
wok place at widely different times between about 1866
and 1905, the most important years being those when
- 48 -
he was at Johns Hopkins. Unlike Jevons, Peirce gave
his approval to Boole's view of the logic of deduction as
having close connections with mathematics, though he
was in agreement with Ievons that any system of such a
logic has to be limited in its use of mathematics by the
need of letting adl its laws and rules and all the steps
which may be taken in it have good sense in logic.
There are four parts of logic that were much helped
to go forward by Peirce : the logic of classes, of relations,
of statement connections and of statements in which
there are variables. These four will be touched on in
turn.
7.5. Peirce's use if the ' a part of ' relations.
Peirce was the first worker in Boole's sort of logic to make use
of the new operation of addition that had been put forward
by De Morgan and Ievons. Down the years from
1867, the year ofhis first papers on Mathematical Logic,
Peirce was responsible for' a number of important
developments and of new systems in the algebra of
classes. Specially to be noted was his joining a new
relatiOn between classes to those which there had been
in Boole's algebra. This new relation was 'a part of',
which is currently marked by the sign C ; the sense of
x C y is that every element of the class x is an element of
the class y. In Boole's algebra thought is able to be
given to this relation, which is one of.the most important
ones in logic and mathematics, only by an equation :
xx(l-'Y)=0. But it is in fact a great help in logic and
mathematics to have a simple sign for this relation.
Further, there is this to be said for Peiroe's use of the
'a part of' relation as a simple relation: this relation in
the algebra of logic is like the relation of 'less than or
equal to' in number algebra and, keeping this in mind,
one may make new discoveries of laws of logic. If the
- 49 -
algebra of classes is, in its form, like the algebra of the
numbers 0 and 1, then, because there are true laws
ming the relation 'less than or equal to' in this number
algebra, there are true laws using the relation 'a part
of' in the class algebra, This was overlooked by
Boole. Lastly, the sign C has the sense in the algebra of
statement connections of material implication (2.8);
when there was a separate sign for this statement connection,
one was more readily able to give it attention
and to make discovery of its properties. Peirce himself
and others gave it much attention. Because of this, it
became the generally used statement connection of the
implication sort almost without a iight, even though
material implication is a somewhat special and strange
sort of implication. Its attractions are that it is the least
strong sort of implication and that it is generally better
than the others for the purposes of mathematics.
- 50 -
7.6. Peirce's theory of relations. Peirce gave a full
account of his theory of relations in his paper 'The
Logic of Relatives' printed in
Studies in Logic by Members
of the Johns Hopkins University (1883). His theory
is based on Boole's algebra and on De Morgan's
Formal
Logic (1847) and papers of De Morgan. De Morgan put
much weight on the idea of relations such as 'to the
right of' which if they are true of the things a and b (in
that order) and are true of the things b and c (in that
order), then they are true every time of the things a and
c (in that order): if a is to the right of b and b is to the
light of c, then a is to the right of c. Further, De Morgan
gave weight to the idea of an operation of joining
two things together in thought so that they have a relation
between them such as that between a head and a
house in what is marked by ' a head of a horse' or between
a daughter and a teacher in what is marked by
' a daughter of a teacher'. (It may be pointed out that
there are in fact a great number of different relations of
this sort, though this is kept dark by the uncertain
connection word ' of '.) It was De Morgan's belief that
the ' of ' relation was in some ways like the operation or
relation marked by x in arithmetic or algebra ; an
example is the number law a x (b+c)=(a x b)+(a x c),
which is, in addition, a law in the logic of relations,
giving x the sense of ' of ' and + the sense of ' or ' : the
father of a teacher or a manager is the father of a
teacher or the father of a manager. However, though
relations were talked about by De Morgan and what he
said had no small effect on Peirce, he did not get so far
as building a systemlor a mathematics of relations; this
is what Peirce made the attempt to do.
In 'The Logic of Relatives' an attempt was made by
Peirce, using the work of Boole and De Morgan, to give
a general theory like a mathematics of things having
relations, these wi]1 be named here 'relation things'.
The relation thing 'father' among persons may be said
simply to be the class of all the groupings (I : J ) where
I is a father and .I is an offspring of I; so (James Mill :
John Stuart Mill) is an element of the class that is the
relation thing ' father ' but (Roger Bacon : Francis Bacon)
is not an element of that class. More generally, any
relation thing r may be said to be the class whose
elements are all the groupings (I :J ) of the things I and
J' such that, as one would commonly say, I is an r of J
(or something near this). One value of this account by
Peirce is its putting an end to the belief that a relation
is something very strange and special. In this way
Peirce did for relations what Boole had done for possible
subjects and predicates' as qualities had been
turned by Boole into classes of the things that have the
qualities, relations were turned by Peirce into classes of
51
the groups of things between which there are the relations.
Another value is in its suggestion that ideas in
mathematics which might be viewed at first as not a
part of logic at all are able to be given
definitions which
nnake use only of what is a part of logic ; mathematics
and logic then are not quite as separate as it seemed,
and it might even be possible for all the ideas of mathematics
to be broken up into ones of logic, mathematics
becoming an expansion of logic and no more than an
expansion of it. (A definition is a statement or rule
saying what sense some word or other sign has or is to
have, or saying that some simple or complex sign is
going to be used as equal to another sign or group of
signs.)
- 52 -
7.7. Peirce's logic of statement connections.
A logic of relations is needed for a logic of mathematics
because there is a great number of relations playing an
important part in mathematics, for example 'square
root of', ' between', ' in the same plane as', ' greater than',
' equal to'. Relations are some of the material of mathematics.
But the logic of statement connections is a
theory of a part of the form, not of the material, of
mathematics. Not that the business of these two sides
of logic is only with mathematics -- far from it. But
much of their development, and that of other sides of
logic, has in fact been caused by the desire of building
a logic that is at least a good one for the needs and
purposes of mathematics. There were chiefly three new
ideas in the logic of statement connections for which
Pdroe was responsible. The first was the idea that it is
possible to give definitions of all the different statement
connections between one or two statements, as ' not',
' and', ' or' and the material implication ' if ... then', by
ming the one connection 'the two are false' or by using
the one connection 'not the two are true'. For example,
' the two are False' being marked by the writing of the
statements s and t side by side so that 's and t are false'
becomes ' st', then ' s is false' becomes ' ss', 's is true or
t is true' becomes ' (st) (st)', and so on.
Secondly, Peirce gave an account of a ' decision process'
for testing if a complex of signs in a system of
statement connections is or is not representative of a
true law of logic. A process is said to be a decision process
if it is fully automatic -- able to be done by machine
-- and gives a test of which complexes of signs in a
system are laws of the system, a complex of signs being
a law if and only if it has a certain property P. In
Peirce's logic of statement connections the decision
process is to give the values 'v and f (from the Latin.
verum = true and falmm = false) in all possible ways
to the variables in the complex of signs and, by definitions,
to get the value of the complex itself worked out
from the values of its parts; if this value is necessarily
v for all the possible values of the variables, then the
complex is a law, while if this value may be f for some
values of the variables, the complex is not a law; so the
property P here is the property of a complex's having
the value v for all possible values of the variables in it.
Peirce makes this decision process clear in relation to
the complex C that is of the form (s->t)->[(t->u)->
(s->u)], where -> is a sign of material implication. By
definition, A->B has the value U whatever values v and
fthe parts A and B of a complex might have but for A
having the value o and B having the valuef, when A->B
has the vaduef: a material implication is false if and only
if the condition is true and the outcome is false. Now
Peirce's argument is this, that if C ever has the value f,
then, from the definition, s->t has the value v and
(t->u)~>(s->u) has the value f; but if this last has the
- 53 -
value f, then, from the defiinition, t->u has the value v
and s->u has the value f; and 80, again from the
definition, s has the value v and u has the value f.
Because u has the value f and t->u has the value v,
necessarily, from the given deinition of material implication,
r has the value But then, because s has the
value v and 11 has the value f, s->t necessarily has the
value f. This, however, is not in harmony with s->t
having the value U when C has the value f (10 lines
back). So it is not possible for C to have the value f at
all' if it is given that value, one gets values that are not
in harmony with the definition of material implication.
In ways like this all complexes of signs in a system of
statement connections may be tested to see if they may
be given the value f without eifecting values that are
not in harmony with some definition, in relation to
v and f, of a statement connection.
Thirdly, Peirce had the idea of building upa theory
of the logic of statement connections as a system based
on axioms. Because the deduction of theorems about
statement connections from axioms is very hard, it was
Peiroe's suggestion that the decision process we were
outlining a minute back be uSed for supporting the
right of those complexes which are unable to have and
to keep the value
f to be theorems in the system and so
representatives of laws of the logic of statement con-
nections. The validity of any argument in mathematics
which is dependent on statement connections is con-
ditioned by these laws: if the argumentS validity is
dependent only on statement connections, then the
argument has validity if and only if it is in agreement
with these laws; if the argument's validity is dependent
on statement Connections and on other sorts of reasoning,
then agreement with those laws is necessary but not
enough to make certain of validity. Peirce's system has
four axioms in which material implication is the only
statement connection and one axiom whose purpose
seems to be to give a definition of the idea of ' not ' or
' false '.
- 54 -
7.8. Peirce's quantifier logic. The statement ' 4 is the square of 2 '
is true or false, as is ' the death of Newton
took place in 1900'. We will say that such statements
are two-valued ones, the values being 'true' and 'faise'.
On the other hand, 'x is the square of y', where the
range of the variables is the class of the numbers 1, 2,
3, ... , and 'the death of x took place in y', where the
range of x is the class of persons and the range of y is
the class of years in the current system, are statements
that are not two-valued ; they are not true and they are
not false. We will say that a statement which is not two-
valued and in which there are variables is a statement
with free variables. s(x), t(x, y), ufx, y, 2) and so on are
used for noting such statements, for example s(x) is
the name of any statement having one variable in it.
(This is a little over-simple; it is unnecessary here to
go into delicate details, because we are having only a
bird's-eye view of logic.)
Looking at the statements 'some x is the father of
some y', 'some x is the father of every y', 'every x is the
father of some y' and 'every x is the father of every y',
the range of the variables being the class of persons,
one sees that all four of them are two-valued; the first is
true and the others are false The first statement says
that someone is somebody's father, the second that
somebody is the father of everyone, the third that
everyone is somebody's father and the fourth that
everyone is the father of everyone. Putting these four
statements in a somewhat different form the iirst be-
comes 'there is an x and diere is a y such that: x is the
- 55 -
father of y', the second becomes 'there is an x such that
for every y: x is the father of y', the third 'for every x
there is a y such that: x is the father of y' andthe
fourth 'for every x and for every y: x is the father Ofy'.
Though the part coming after the : in these four state-
ments is a statement with free variables, the complete
statements themselves are two-valued, 'There is a' and
'for every' are named quaotyiers, the first being the
'existence' and the second the 'general' quantiiier, »
which were noted by Peirce by the Greek letters Z
andII.
Peirce gave much attention to the logic of variables,
this being that field of logic whose business is with
implications and arguments some or all of whose con-
ditions or outcomes are statements with free variables
or that have quantifiers in them. A great number of
theorems and arguments in mathematics are dependent
on the use of quantifiers; so this part of logic is very
important. With Frege but having no knowledge of
Frege's work, Peirce was the first to be interested and
to make deep discoveries in this held. Using - as the
sign of 'not' or 'it is false that', some simple examples
of laws given by Peirce are :
(1)~<2»»(==)=(H==)-(»),
(2) -(Hx)(x)=(zx)-_¢(x),
(3) (L'x)(5(x)+¢(x))=((L'x)¢(x))+((_2')z(x)),
(4) (H»°=)(#(x)+£{x))=((IT4¢}f(x))+({Ux)¢(x))»
(5) <2»>(flyw(==, y)~>(1Iy)(-'5`x)"(x» y),
and laws which are the same as (3) and (4) but in which
x takes the place of +.
- 56 -
7.9. Schroeder's algebra of logic.
Ernst Schroeder's (1841 - 1902) two chief books on Mathematical
Logic were
Der Operationshreis der Logikkalkus
('The Range of Operations of the Mathematics of Logic') and
Vorlesungen über die Algebra der Logik (' Teachings on the Algebra of Logic').
The first of these two books (1877)
is very short, being only 37 pages long. There is in it a
simple system of the algebra of logic, this system being
based on Boole's though different from it in some important
points where the discovery had been made by
later thought and experience that changes were wise or
necessary. The tendency of these changes was in the
same direction as that of those changes that had been
put forward by Ievons and Peirce: the algebra is to be
logic in the dress of mathematics, not a bit of mathematics
which almost by accident is open, completely or
in part, to being viewed as a theory of logic.
The writing of the Vorlesungen was in four parts that
came separately on the market between the years 1890
and 1905. This book gave a full account of the current
Mathematical Logic in so far as this had its roots in
algebra, in it were discussions and systems going into
the smallest details of the logic of classes, of statement
connections and of relations.
Much that Schroeder did was new, though often
resting on older material. Among the number of his
theorems that were of special value was that to the
effect that all the laws of the algebra of logic may be
grouped in twos, it being possible from a knowledge of
One of the two to get straight away, by a simple rule of
exchange, a knowledge of the other one. The ruie is
that, in the given law, + and x , and 1 and 0 (the signs
of All and Nothing), be put in one another's places; for
example, from the law x x 1=x one may get the law
x+0=x, from the law x+-x=1 one may get the law
xx-x=0 and from the law xx(y+z)=(xxy)+(xxz)
one may get the law x (yx z)={x y)x (#=+#) This last
law was first given by Peirce; it is not a law in Boole's
- 57 -
algebra because of the different use Boole makes of +-
Again, the laws x+x=x, x+yIy+x, (x+y)+z=x+
(y-I-af) and x=x+(x xy) and the four laws that may be
got from these by the rule of exchange were seen by
Schroeder to be of a somewhat separate sort from the
other laws, In fact, these 8 laws are very important for
Lattice Theory (see end of 7.1) because a class L is a
lattice if and only if these laws are true in it.
Not to be confused with Schrödinger's cat, a paradox devised by Erwin Schrödinger in1935.
7.10. Whitehead's and Huntington's work on the algebra of logic. The present history of the algebra of
logic will be ended now by our saying a word or two
about the stage in the development of this algebra that
came 'm the ten years after its first 50 years, 18471891
This stage is marked by two things. Firstly, by Whitehead's
(1861-1947) attempt in his
Universal Algebra
(1898) to give a completely general theory of algebra
and to say what bodies of axioms and definitions will
be good starting points for the different systems covering
this part of mathematics ; this book was the first
after
The Laws of Thought to take seriously the algebra
of logic as a field of everyday mathematics and it was
the Erst to make clear the connections between this
branch and other branches of Abstract Algebra.
Secondly, this stage is marked by the Erst conscious
work in the metamathematics of the algebra of logic (see
51). Using tricks and processes of a sort designed by
Plano and by others of the Italian school, Huntington
(1874-1952) gave a
demonstration -- a good argument
in support of what is said to be true -- of the consistency
of a number of different axiom systems of the algebra
of logic, the substance of these systems being copies of
ones in the books of Schroeder and Whitehead, and he
gave a demonstration that every axiom of his systems is
unable to be got as a theorem from the rest of the
axioms of the same system, so no axiom of a system is
dependent on`the others. This bit of second-order
theory of the algebra of logic -- the logic of logic -- was
done at the time (1904), between 1890 and 1905, when
second-order theories of mathematics --- metamathematics
-- were starting to be seen as interesting and as a
very important part of the theory of the bases of
mathematics .
- 58 -
8 FREGE'S LOGIC
8.1 . The mathematics of logic and the logic of mathematics.
Mathematical Logic has two sides.
One, in its roots in the past (we say this because all
sciences have a tendency to become self-conscious and,
if necessary for their best development, to let the
material that was their first business get dropped from
view), is the science of the deduction form of reasoning,
this science using ideas and operations like those in
mathematics and copying mathematics in being system-
making. This side, then, is the mathematics of logic. It
was in this that Boole and others interested in the
algebra of logic were working. This side has two levels :
the logic of deduction itself in the form of systems
covering statement connections, classes, relations and
statements with variables, and the second-order
theories of those systems, whose birth came nearly 50
years after that of those first-order systems. The other
side of Mathematical Logic is the logic of mathematics.
Here there are two chief sorts of questions to which one
- 59 -
has given attention, certain points of these being in the
same line of direction. One sort of question is that to be
answered by the metamathematics of any given branch
of mathematics--of arithmetic, Euclid's geometry,
group theory or whatever it may be. The second sort of
question is about the bases of mathematics; for example,
is it possible to have a system with the property
of consistency in which the ideas used are enough and
have the right properties so that, firstly, definitions of
all the ideas of the common mathematics may be given
from these and only these ideas, and, secondly, all the
theorems in the common mathematics may be got as
theorems from that system stretched to have new
definitions resting on the ideas in the old ones ? From
the time of Schroeder's death (1902) till now, most
work in Mathematical Logic has been in connection
with the logic of mathematics, and in 8.2-12.6 we will
say something about this work. The reading of this
coming discussion will have to be done very slowly if
the material is going to be taken in. Because we have
little space and because it is possible to get a clear view
of what has been produced by the newer work in
Mathematical Logic only if one goes through the long
stretch of expert details, we will be limiting ourselves
to outlining some of the chief directions which the
later developments in Mathematical Logic have had.
- 60 -
8.2 Reasons why Frege's writings were undervalued.
Gottlob Frege (1848-1925) was Professor of
Mathematics at the University of jena and the writer
of books and papers on logic and the logic of arithmetic,
and on the philosophy of these fields. For a number of
reasons Frege's writings were undervalued till Russell's
Principles of Mathematics (1903) gave an account of
them. One reason was their use of signs for statement
connections the writing of which was not in the normal
direction of left to right but up and down; this makes
the reading of sign complexes much harder. Another
reason was that, generally, those persons whose business
was philosophy were controlled by the impulse not
to go near, still less to go through, a mass of sign complexes
when the purpose of this mass was to be the
support of a theory in philosophy -- here, in the philosophy
of mathematics. Their belief was (and is) that an
argument for or against any such theory has to be made
in everyday language because the value of the theory is
dependent on its starting points, and the opinions which
are at the back of these, being true, and that it is not
possible for these to be supported by anything inside a
mathematics-like structure but only by arguments outside t
he structure. Another reason was that Frege's
theory was so surprising and so far from probable that
there seemed little chance of the trouble of working
through the details of the deductions offered for it
being rewarding. One last reason we would put forward
is that what Frege was doing was off the straight and
narrow road: it was not quite normal mathematics, it
was not quite normal philosophy and it was not quite
normal logic. It was not normal mathematics: it was
the bases of mathematics. It was not normal philosophy :
it was the philosophy of mathematics done by someone
having a first-hand knowledge of mathematics. And it
was not normal logic: it was not the algebra of logic.
(The 'normal' is in relation to the time 1875-1900.)
- 61 -
8.3 Frege's view of the relation between logic and
arithmetic. What was Frege's theory?
It was that
number is an idea (not dependent on the mind for its
existence) which may be broken up into ideas of logic
and which is open to a complete definition by using no
- 61 -
more than these ideas ; and so, further, that arithmetic
itself is a part of logic. In agreement with Frege,
Russell said that arithmetic ' is only a development, without
new axioms, of a certain branch of general logic'. The
journey to the full, tightly reasoned demonstration of
this theory was started by Frege in his
Begriffsschrift ;
eine der arithmetischen nachgebildete Formelsprache des
rm|en Denkens ('Idea-Writing; a sign language, copying
arithmetic, of thought as such') of 1879 and came
to its end with the two parts of his
Die Grundgesetze
der Afithmetik, begrzfsschriftlich abgeleitet ('The Root
Laws of Arithmetic, got by deductions in idea-
writing') of 1893 and 1903. In the
Begriffsschrift Frege
gives an axiom system of the logic of statement connections
and, using the general quantifer (by which, together with
' not ', he gives a definition of the existence
quantifier), he goes deeply into the logic of statements
with variables. In addition, there is an account of some
parts of arithmetic worked out from definitions based
on what are taken to be ideas of logic, for example the
ideas of class, class element and relation, In the Grund-
gesetze there is a complete account of arithmetic, in so
far as this is the theory of the laws of the operations on
numbers of the form p|q where the range of q is
+/-l, +/-2, +/-3, ... and the range of P is the same
together with 0. The heart of Frege's arithmetic is his
definition of a
cardinal number. The simplest cardinal
numbers are 0, 1, 2, 3, ... For Frege, a cardinal number
is a property of a class. To the question, 'What is the
number of men in the Russian army now' the right
answer will be a cardinal number and this number will
be a property of the class 'man in the Russian army
now'. By a 'class' here Frege, unlike Russell, has in
mind the class-idea (that is, the class as an idea), not
the list of things coming under the class-idea. So a
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cardinal number is a property of a class-idea, for example
of the class-idea ' man in the Russian army now '.
Flege's definition of the cardinal number of a class x is
that it is the class of all the classes y such that x and y
have a one-one relation between them; by definition,
classes x and y have a one-one relation between them if
and only if (i) each element of x may be grouped with
some element of y, (ii) each element of y may be
grouped with some element of x, (iii) if any a1 and a2 of
x are grouped with a certain b of y, then a1 and a2 are
the very same element of x, and (iv) if a certain a of x is
grouped with bl and with b., of y, then b1 and B2, are the
very same element of y. This may seem harder than it
is; it is put like this for the purpose of stopping anyone
from being able to say that the definition goes round in
a circle, because it makes use of the cardinal number
one (in 'one-one relation'). What the definition says is
simply this, that x and y have a one-one relation between
them if and only if it is possible for there to be a
body R of groupings (a : b) such that every a of x is the
first name in some grouping (a : b) of R while no a of x
is the first name in more than one such grouping and
every b of y is the second name in some grouping (a : b)
of R while no b of y is the second name in more than
one such grouping. From his definition of cardinal number
and by the instruments of logic Frege gets the theorems
of everyday arithmetic at the end of very long chains
of deduction in the form of sign complexes; so even if
one has doubts of his theory being true, at any rate his
industry and expert invention are to be respected.
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8.4 Frege on functions.
One of the most important
ideas at the base of mathematics is that of a function.
The definitions given of this idea in Frege's time did
not, when looked at with care, make very good sense
though they had seemed to make good sense and,
further, they were not in harmony with the uses made
of the idea. As frequently in philosophy, Frege had a
sharp tongue when attacking, specially when attacking
an unsafe position.
What is a function ? In Frege's time the common
answer to this question was to say, as the German
writer on mathematics, Ernst Czuber, did : 'If every
value of the number variable x that is an element of the
range of x is joined by some fixed relation to a certain
number y, then, generally, y, like x, comes under the
definition of a variable and y is said to be a function of
the number variable x. The relation between x and y is
marked by an equation of the form y f(x).' (Number
variables are not necessarily limited to the range
0, :l:1, :t2, i3, . . ., or even to numbers of the
sort p/q where P and q are from that range (q-=0):
they may, for example, be variables of numbers like
~/2 or *s/-25.) This answer was soon undermined by
Frege's blows, one after another, from the gun of his
quick-firing brain. 'One may straight away take note of
the fact', says Frege, 'that y is named a certain number,
while, on the other band, being a variable, it would
have to be an uncertain number. y is not a certain and
it is not an uncertain number; but the sign 'y' is
wrongly placed in connection with a class of numbers
[the range of the variable y] and then it is talked about
as if there were only one number in this class, It would
be simpler and clearer to say: "With every number of
an x-range there is some fixed relation by which it is
joined to another [not necessarily different] number.
The class of all these last numbers will be named the
y-range." Here we certainly have a y-range, but we
have no y of which one might say that it was a function
of the number variable x.
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' Now the limiting of the range does not seem to have
anything to do with the question of what the function
itself is. Why not take as the range the class of all the
numbers of arithmetic or the class of all the complex
numbers (of which the first class is a part)? The heart
of the business is in fact in quite a different place which
is kept out of view in the words "joined by some fixed
relation". What is the test of the number 5 being joined
by some fixed relation to the number 4? The question
is unable to he answered if it is not somehow made
more complete. With Herr Czuber's account it seems
as if, for any two numbers, the decision was made at
the start that the first is, or is not, joined by the relation
to the second. Happily Herr Czuber goes on to say :
"My definition makes no statement about the law controlling
the joining by the relation; this law is marked in
its most general form by the letterf but there is no end
to the number of different ways in which it may he
fixed." Joining by a relation, then, takes place in agreement
with some law and it is possible, by the use of
thought, for different laws of this sort to be produced.
But now the group of words "y is a function of x" has no
sense, there is need of its being made complete by the
addition of the law. This is an error in the definition.
And, without doubt, the law, which is overlooked by
this definition, is truly the chief thing. We see now that
what is variable [changing] has gone completely out of
view and in its place the general has come into view,
because that is what is pointed at by the word "law".'
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8.5 Functions as rules. What Frege says about
functions in the paper from which we have taken these
lines has become normal in the present-day account of
the idea of a function. A function f is a rule about
classes X and T, X being named the class of definition
off and T the range off, the rule producing a class of
ordered groupings (x : y) where x (which is named an
'argument' of f) is an element of X and y (which is
named the 'va1ue' of f for x) is that selection from
among the elements of T which is made in agreement
with the rule ; the class of all the selections y from T is
named the class of values off For example, if the class
of definition of f is the class X of all the numbers 1, 2,
3, . . ., the range of f1 is the same class and f is the rule ' To
every element x of X let 2x be joined', then the class of
ordered groupings produced by this rule has (1 : 2),
(2 :4), (3 : 6), . . . as its elements, so the class of values of
f has 2, 4, 6, . . . as its elements , again, if the class of
definition of f, is the class X of all the numbers -1, -2,
-3, . . ., the range of j, is the class of all the numbers
1, 2, 3, . . . and fl is the rule ' To every element x of X
let x2 be joined ', then the class of ordered groupings
produced by this rule has (-1 : 1), (-2 : 4), (-3 : 9), . . .
as its elements and the class of values off, has 1, 4, 9, . . .
as its elements .
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9 CANTOR'S ARITHMETIC OF CLASSES
9.1 Finite and enumerable classes.
A short way of writing that a one-one relation is possible between any
two classes x and y is 'x ~= y'. If x ~= y, then by a definition
of Cantor's (1845-1918), x and y are said to have
the same cardinal number or the same power, the sign
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for this fact being the equation'§=§'. (This definition
is clearly like Frege's definition of ' cardinal number';
but Cantor does not give a definition of ' cardinal number',
only of ' having the same cardinal number'. Cantor
and Frege were working at the same time, but quite
separately; one was not copying the other at all.) If
N,,1 is the class of the 'natural' numbers from 1 through
m, so the elements of it are 1, 2, 3, ... , m--1, m, then a
class x is finite if and only if it has no elements or, for
some natural number m, x- N,,,. If N is the class of
all the natural numbers, its elements being 1, 2, 3, , ..
where '.. ,' is used for marking that the list goes on for
ever, then a class x is said to be
enumerable if and only
if x ~= N. It is strange how great a number of quite
different classes are enumerable. For example, Cantor
gave demonstrations that the class Z of all the numbers
0, il, :t2, :l;3, ... is enumerable; the class of all the
numbers p/q greater than 0 and less than 1., where P and
q are elements of N, is enumerable; but what to the
mind's eye would be seen as a very much greater class,
the class Q of all the numbers p/q, where P and q are
elements of Z (q-=0), is again enumerable ; and, further,
the class of all numbers which are roots of an equation
of the form a,,w"+a"_1w"-1+... +a1w+a0,0, where
each "4 is an element of Q, is enumerable.
In view of these surprising facts it is natural to put
the question: Are there any classes that are unenumerable,
that is which are not finite and which are unable to
he placed in a one-one relation with N? The answer is,
Yes. The class I of all the numbers of the form 0-a1a,a:
... an ..., where the a's have values in the range 0, 1, 2,
..., 9, is an unenumerable class, it has, for example,
0-127845 '7 023640000 4 l I and 0~765976 l n l as
elements. There are other unenumerable classes.
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9.2 Cantor's cardinal arithmetic. Though Cantor
did not give a definition of 'cardinal number' he made
me of cardinal numbers as things having some sort of
existence and having properties and relations among
themselves, in much the same way as one makes use of
the numbers 1, 2, 3, ... in everyday arithmetic without
definitions of them. By definition, the cardinal
number of a class x is greater than the cardinal number
of a class y -if >§-if and only if there is some part
x1 of x such that X1 is not all of x, xlzy and there is no
part y1 of y such that yl ~= x, for example, N, >N,
and N> Nina- Cantor gave a demonstration of a general
theorem which says that, for any class y having
elements, the class Uy whose elements are all the
classes x such that x Cy (7.5) has a greater cardinal
number than y has: Uy >y. An outcome of this
theorem is that there are ever greater and greater
cardinal numbers, if only because y < Uy < UUy
< UUUy < UUUUy < ...
An arithmetic of cardinal numbers was worked out
by Cantor, definitions of the operations of addition and
so on for cardinal numbers being given and the discovery
and demonstration made of laws of the arithmetic.
To take the simplest operation, that of addition :
by definition, if x and y are any two classes such that no
element of the one is an element of the other, then
219 = 3:+5', where the + sign in the left-hand side of
the equation is of the addition of classes (x +y being the
class whose elements are all the things that are elements
(of x or of y). The common laws of addition are true for
cardiinal numbers ; if k1, k, and k, are cardinal numbers,
k,+k,==k,+k, and (k,+k,)+k,=k1+(k,+k,). On the
other hand, there are, further, some uncommon laws ;
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for example, if m is the cardinal number of any finite
class and k is the cardinal number of N, then the
addition of m k's is equal to only k (mk=k) and
km=k.
9.3 Cantor's ordinal arithmetic.
The arithmetic of
still another sort of number, that named cardinal, was
worked out for the first time by Cantor. Ordinal numbers
are based on well-ordered classes, that is on classes
which are made to have a certain order among their
elements so that there is a first element, a second
element, and so on. For finite classes ordinal and cardinal
numbers are the same; however, for classes that
are not finite the numbers are not the same because it is
possible to make all such classes have widely different
order relations. And, as for cardinals, there is no limit
to the number of higher and higher ordinals.
The theory of cardinal and ordinal numbers -- the
arithmetic of classes -- has been one of the most fertile
inventions of the newer mathematics and is one of the
most important branches of Mathematical Logic. The
details of this theory are highly complex, and we are
unable to go into them here.
9.4 Burali-Forti's and Russell's discoveries. But
the warm bright days when the sunlight was playing
on the flowers in Cantor's garden and all seemed well
were sadly soon over. Black clouds came up in the sky
and the earth was made dark. Men of mathematics were
put in doubt, fearing to see the downfall and destruction
of one of the beautiful theories of mathematics.
The cause of the trouble was the discovery that
Cantor's arithmetic of cardinal and ordinal numbers
did not have the needed property of consistency (5.1).
In 1897 Burali-Forti (1861-1931) gave a demonstration
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that the ordina1 number of the well-ordered class W of
all the ordinal numbers is not an element of that class ;
so, on the one hand, all ordinal numbers are, by
definition, elements of W while, on the other hand,
not all ordinal numbers are elements of W because
the ordinal number of W itself is not an element
of W. Again, in 1901 Russell (1872-) saw the loss of
consistency of cardinal arithmetic in connection with
the most general class A. By definition, A is the class
whose elements are all the things which are not classes,
together with all the classes of those things, together
with all the classes of classes of those things, and so on.
A, then, is a class such that all its parts are elements of
it. Because of this fact, it is not possible for there to be
more parts of A that there are elements of A, that is
MA (the cardinal number of the class of all the
parts of A is less than or equal to the cardinal number
of A). However, this is not in agreement with Cantor's
general theorem (9.2) which makes it necessary here
that the cardinal number of UA is not less than or
equal to, but greater than, the cardinal number of A.
From the discovery of this example Russell got the idea
of another one which is the most noted of all such examples.
He sent it to Frege at the very time that the
second part of Frege's
Grundgesetze der Afithmetik was
coming near the end of its printing before being offered
to the public ; Frege was made conscious of the fact
that his system of classes, like Cantor's, did not have
the needed property of consistency. Russe1l's new idea
was this : without there being any suggestion that any
class is able to be an element of itself, let E, by definition,
be the class of all classes that are not elements of
themselves , for example, the class ' man' is an element of
E because ' man' is not an element of itself, this class
having only men, and no classes, as elements. Is E an
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element of E ? If (I) E is an element of E, then (II) F is
not an element of E because the only elements of E are
those classes which are not elements of themselves, on
the other hand, if (II) E is not an element of E, then
(I) E is an element of E because of the definition of E.
By a common law of logic one or the other of (I) and
(II) is true. But whichever one is true makes the other
one true in addition. So the theory of classes has two
Opposite theorems in it : E is an element of E ; F is not
an element of E.
The effect of these discoveries on the development of
Mathematical Logic has been very great. The fear that
the current systems of mathematics might not have
consistency has been chiefly responsible for the change
in the direction of Mathematical Logic towards metamathematics,
for the purpose of becoming free from
the disease of doubting if mathematics is resting on a
solid base. A special reason for being troubled is that
the theory of classes is used in all other parts of mathematics,
so if it is wrong in some way, they are possibly
in error. Further, quite separately from the theory of
classes, might not discoveries of opposite theorems in
algebra, geometry or Mathematical Analysis suddenly
come into view, as the discoveries of Burali-Forti and
Russell had done ? It has been seen that common sense
is not good enough as a lighthouse for keeping one safe
from being broken against the overhanging slopes of
sharp logic. To become certain with good reason that the
systems of mathematics are all right it is necessary for
the details of their structures to be looked at with care
and for demonstrations to be given that, with those
structures, consistency is present.
This last point and the fears and troubled mind we
have been talking about in these lines have been and are
common among workers in what is named 'the bases of
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mathematics', that is axiom systems of logic-classes-and-arithmetic.
However, some persons, with whom the
present writer is in agreement, have a different opinion.
They would say that the well-being of mathematics is
not dependent on its 'bases'. The value of mathematics
is in the fruits of its branches more than in its 'roots';
in the great number of surprising and interesting
theorems of algebra, analysis, geometry, topology,
theory of numbers and theory of chances more than in
attempts to get a bit of arithmetic or topology as simply
a development of logic itself. They would say that the
name 'bases of mathematics' is a bad one in so far as it
sends a wrong picture into one's mind of the relations
between logic and higher mathematics. Higher mathematics
is not resting on logic or formed from logic.
They would say that the troubles in the theory of
classes came from most special examples of classes, and
such classes are not used in higher mathematics. They
would say further that though to be certain of consistency
is to be desired if such certain knowledge is
possible to us, a knowledge of the consistency of the
theories of mathematics that is probable is generally
enough and the only sort of knowledge of consistency
that one does in fact generally have. And they would
say that such probable knowledge is well supported if
the theories of mathematics have been worked out much
and opposite theorems in them have not come to light.
There is no suggestion in all this that Mathematical
Logic is not an important part of mathematics; the
view put forward is that there is much more to mathematics
than Mathematical Logic is and might ever
become. And there is no suggestion that the question
of consistency and like questions, and the discovery of
ways of answering them, are hot important; to no small
degree Mathematical Logic now is as interesting and
important as it is because of its interest in such questions and answers.